In the Multiple Regression Model with Two Explanatory Variables $$Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { 1 i } + \beta _ { 2 } X _ { 2 i } + u _ { i }$$

question 25

Essay

In the multiple regression model with two explanatory variables $$Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { 1 i } + \beta _ { 2 } X _ { 2 i } + u _ { i }$$ the OLS estimators for the three parameters are as follows (small letters refer to deviations from means as in $z _ { i } = Z _ { i } - \bar { Z }$ ):
$$\begin{array} { c } \hat { \beta } _ { 0 } = \bar { Y } - \hat { \beta } _ { 1 } \bar { X } _ { 1 } - \hat { \beta } _ { 2 } \bar { X } _ { 2 } \\\hat { \beta } _ { 1 } = \frac { \sum _ { i = 1 } ^ { n } y _ { i } x _ { 1 i } \sum _ { i = 1 } ^ { n } x _ { 2 i } ^ { 2 } - \sum _ { i = 1 } ^ { n } y _ { i } x _ { 2 i } \sum _ { i = 1 } ^ { n } x _ { 1 i } x _ { 2 i } } { \sum _ { i = 1 } ^ { n } x _ { 1 i } ^ { 2 } \sum _ { i = 1 } ^ { n } x _ { 2 i } ^ { 2 } - \left( \sum _ { i = 1 } ^ { n } x _ { 1 i } x _ { 2 i } \right) ^ { 2 } } \\\\\hat { \beta } _ { 2 } = \frac { \sum _ { i = 1 } ^ { n } y _ { i } x _ { 2 i } \sum _ { i = 1 } ^ { n } x _ { 1 i } ^ { 2 } - \sum _ { i = 1 } ^ { n } y _ { i } x _ { 1 i } \sum _ { i = 1 } ^ { n } x _ { 1 i } x _ { 2 i } } { \sum _ { i = 1 } ^ { n } x _ { 1 i } ^ { 2 } \sum _ { i = 1 } ^ { n } x _ { 2 i } ^ { 2 } - \left( \sum _ { i = 1 } ^ { n } x _ { 1 i } x _ { 2 i } \right) ^ { 2 } }\end{array}$$ You have collected data for 104 countries of the world from the Penn World Tables and want to estimate the effect of the population growth rate $\left( X _ { 1 i } \right)$ and the saving rate $\left( X _ { 2 i } \right)$ (average investment share of GDP from 1980 to 1990) on GDP per worker (relative to the U.S.) in 1990. The various sums needed to calculate the OLS estimates are given below: $$\begin{array} { l } \sum _ { i = 1 } ^ { n } Y _ { i } = 33.33 ; \sum _ { i = 1 } ^ { n } X _ { 1 i } = 2.025 ; \sum _ { i = 1 } ^ { n } X _ { 2 i } = 17.313 \\\\\sum _ { i = 1 } ^ { n } y _ { i } ^ { 2 } = 8.3103 ; \sum _ { i = 1 } ^ { n } x _ { 1 i } ^ { 2 } = .0122 ; \sum _ { i = 1 } ^ { n } x _ { 2 i } ^ { 2 } = 0.6422\end{array}$$ $$\sum _ { i = 1 } ^ { n } y _ { i } x _ { 1 i } = - 0.2304 ; \sum _ { i = 1 } ^ { n } y _ { i } x _ { 2 i } = 1.5676 ; \sum _ { i = 1 } ^ { n } x _ { 1 i } x _ { 2 i } = - 0.0520$$ (a)What are your expected signs for the regression coefficient? Calculate the coefficients
and see if their signs correspond to your intuition.

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